Fluid extraction from porous media by a slender permeable prolate-spheroid

Abstract

a b s t r a c tFluid extraction from porous media by a slender permeable prolate-spheroid is analyzed. It is found that the flux density along the spheroid-reservoir surface increases with the sharpness of the spheroid tip; and the fluid production rate is significantly higher than that predicted by the cylinder model used in the literature. It is shown that the flow in the reservoir is a superposition of a confocal flow and a redistributive flow, with only the confocal flow being productive. For high spheroid-to-reservoir permeability ratios, there exists an optimum length-to-radius ratio that maximizes fluid production rate for a given spheroid body volume.

Minimizing hydraulic resistance of a plant root by shape optimization

“…Furthermore, for any given permeability ratio, there is a minimum resistance at an optimum value of length-to-radius ratio 1 l . As discussed previously, an increase in 1 l causes a corresponding decrease in 0 ξ and 1 ξ , which increases the 'slenderness' of the spheroid and enhances the tip singularity (Chen, 2015 andChen, 2016), while decreasing the area available for water uptake at the base. While the tip singularity is present for all values of λ , it is important to note that the limitation of a constant λ is imposed and not one of constant sD C .…”

“…This configuration yielded an analytical solution in terms of Legendre polynomials and the resulting flow field is three-dimensional near the tip of the root and shows a converging flow field close to the tip. This work also showed that by not considering the finite length of the root and the resulting three dimensional flow, the Gardner (1960) model under-predicts the flow rate in comparison to the model proposed by Chen (2015). The physics of the resulting flow field is discussed in detail by Chen (2015).…”

“…This decrease of 0 ξ has two consequences for the flow rate at the base of the stele. It increases the inverse square root singularity of the pressure gradient at the tip of the stele (Chen, 2015 andChen, 2016), which significantly increases the velocity of the flow in stele. On the other hand, it also causes a reduction in ξ − according to (132).…”

“…The resistance is then calculated as the ratio of the pressure difference between the root-soil surface pressure and the pressure at the base of the stele, to the volumetric flow rate in the stele. In the composite root model of Chen (2016), the flux from the cortex to the stele has a singularity close to the tip of the stele and this is due to the same physics that cause the flow from the soil to a single root to have a singularity at the root tip, as discussed by Chen (2015).…”

“…To account for this three dimensionality of the flow, Chen (2015) modelled the root as a slender semi-prolate spheroid consisting of a single uniform structure, the stele. This configuration yielded an analytical solution in terms of Legendre polynomials and the resulting flow field is three-dimensional near the tip of the root and shows a converging flow field close to the tip.…”